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Properties

Label 2-3-3.2-c64-0-17
Degree $2$
Conductor $3$
Sign $-0.847 + 0.531i$
Analytic cond. $77.8210$
Root an. cond. $8.82162$
Motivic weight $64$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.95e9i·2-s + (−1.56e15 + 9.84e14i)3-s + 1.46e19·4-s − 1.66e22i·5-s + (1.92e24 + 3.06e24i)6-s + 1.45e27·7-s − 6.46e28i·8-s + (1.49e30 − 3.09e30i)9-s − 3.25e31·10-s − 3.23e33i·11-s + (−2.29e34 + 1.44e34i)12-s − 4.67e35·13-s − 2.84e36i·14-s + (1.64e37 + 2.61e37i)15-s + 1.43e38·16-s − 2.96e39i·17-s + ⋯
L(s)  = 1  − 0.454i·2-s + (−0.847 + 0.531i)3-s + 0.793·4-s − 0.716i·5-s + (0.241 + 0.385i)6-s + 1.31·7-s − 0.815i·8-s + (0.435 − 0.900i)9-s − 0.325·10-s − 1.53i·11-s + (−0.671 + 0.421i)12-s − 1.05·13-s − 0.600i·14-s + (0.380 + 0.606i)15-s + 0.422·16-s − 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(65-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+32) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.847 + 0.531i$
Analytic conductor: \(77.8210\)
Root analytic conductor: \(8.82162\)
Motivic weight: \(64\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :32),\ -0.847 + 0.531i)\)

Particular Values

\(L(\frac{65}{2})\) \(\approx\) \(1.918358500\)
\(L(\frac12)\) \(\approx\) \(1.918358500\)
\(L(33)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.56e15 - 9.84e14i)T \)
good2 \( 1 + 1.95e9iT - 1.84e19T^{2} \)
5 \( 1 + 1.66e22iT - 5.42e44T^{2} \)
7 \( 1 - 1.45e27T + 1.21e54T^{2} \)
11 \( 1 + 3.23e33iT - 4.45e66T^{2} \)
13 \( 1 + 4.67e35T + 1.96e71T^{2} \)
17 \( 1 + 2.96e39iT - 5.60e78T^{2} \)
19 \( 1 - 2.71e40T + 6.92e81T^{2} \)
23 \( 1 + 3.52e43iT - 1.41e87T^{2} \)
29 \( 1 - 1.02e47iT - 3.92e93T^{2} \)
31 \( 1 + 3.49e47T + 2.79e95T^{2} \)
37 \( 1 + 1.01e50T + 2.31e100T^{2} \)
41 \( 1 - 2.25e51iT - 1.65e103T^{2} \)
43 \( 1 - 2.93e52T + 3.48e104T^{2} \)
47 \( 1 + 2.90e53iT - 1.03e107T^{2} \)
53 \( 1 - 1.39e55iT - 2.25e110T^{2} \)
59 \( 1 + 7.35e55iT - 2.16e113T^{2} \)
61 \( 1 + 7.75e56T + 1.82e114T^{2} \)
67 \( 1 + 4.41e58T + 7.39e116T^{2} \)
71 \( 1 - 4.66e58iT - 3.02e118T^{2} \)
73 \( 1 - 2.64e58T + 1.78e119T^{2} \)
79 \( 1 + 9.87e60T + 2.80e121T^{2} \)
83 \( 1 - 1.45e61iT - 6.62e122T^{2} \)
89 \( 1 + 2.35e62iT - 5.76e124T^{2} \)
97 \( 1 + 1.39e63T + 1.42e127T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.38918346252431951511152591796, −11.44511195992493067569502684969, −10.61148964297154175793935185287, −8.929396991952564339298772243793, −7.21981297864339143129630123635, −5.56864258842351230740950991495, −4.64505658859729867079024277622, −2.98225252907235463294533015798, −1.34380371420616141785707713037, −0.48556477254149072868800816286, 1.58350212947982379267371633952, 2.25827195072067472549941620119, 4.63073438607202482982018746282, 5.82857292145894217651061943870, 7.20336320974532561028329446956, 7.69590052554240593552668518272, 10.31677445595229212833336944417, 11.37398615511539273343844447852, 12.37505943465955071242762868265, 14.48131335321894432877913929560

Graph of the $Z$-function along the critical line